Curve sketching

Curve sketching

In this section we will expand our knowledge on the connection between derivatives and the shape of a graph. By following the "5-Steps Approach", we will quantify the characteristics of the function with application of derivatives, which will enable us to sketch the graph of a function.

Lessons

Guidelines for Curve Sketchinga) domainb) Intercepts
y-intercept: set x=0 and evaluate y.
x-intercept: set y=0 and solve for x. (skip this step if the equation is difficult to solve)
c) Asymptotesvertical asymptotes: for rational functions, vertical asymptotes can be located by equating the denominator to 0 after canceling any common factors.
horizontal asymptotes: evaluate limx→∞f(x)lim_{x \to \infty } f(x)limx→∞​f(x) to determine the right-end behavior;
evaluate limx→−∞f(x)lim_{x \to -\infty } f(x)limx→−∞​f(x) to determine the left-end behavior.
slant asymptotes: for rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator.
d) Computef′(x) f' (x)f′(x)
find the critical numbers:
• use the First Derivative Test to find: intervals of increase/decrease and local extrema.e) Computef′′(x) f'' (x)f′′(x) • inflection points occur where the direction of concavity changes.
find possible inflection points by equating thef′′(x) f'' (x)f′′(x) to 0.
•Concavity Test:
•inflection points occur where the direction of concavity changes.

1.

Use the guidelines to sketch the graph of:
f(x)=x3−8x3+8f(x)=\frac{x^3-8}{x^3+8}f(x)=x3+8x3−8​